Essentials in the theory of framed structures . for any reason the M-diagram had not been symmetrical,the ordinates at B and D could not have been assumed equal;and the tangent to the elastic curve through C would not havebeen horizontal. There would have been three unknownordinates, M2 at B, M3 at C and if 4 at D; and three elasticequations would have been necessary for a solution. Thesethree equations may be written as follows: (a) draw a tangent Sec. n RESTRAINED AND CONTINUOUS BEAMS 267 to the elastic curve at B and establish a relation between thetangential deviations at A and C; (b) draw
Essentials in the theory of framed structures . for any reason the M-diagram had not been symmetrical,the ordinates at B and D could not have been assumed equal;and the tangent to the elastic curve through C would not havebeen horizontal. There would have been three unknownordinates, M2 at B, M3 at C and if 4 at D; and three elasticequations would have been necessary for a solution. Thesethree equations may be written as follows: (a) draw a tangent Sec. n RESTRAINED AND CONTINUOUS BEAMS 267 to the elastic curve at B and establish a relation between thetangential deviations at A and C; (b) draw a tangent to theelastic curve at C and establish a relation between the tangen-tial deviations at B and D; (c) draw a tangent to the elasticcurve at D and establish a relation between the tangentialdeviations at C and three equations may also be written as follows:Let the tangent to the elastic curve be drawn through B;and let h, h, h and i^ represent the tangential deviations atA, C, D and E respectively. Then establish a relation be-. FlG. 166. tween h and t^, a second relation between h and ti, and a thirdrelation between h and t^. In case the beam is restrained at each end by being fixedin a wall; the M-diagram presents five unknown ordinates, orone at each point of support. This condition requires twoadditional elastic equations, or five in all. These two equa-tions may easily be written, since the tangents to the elasticcurve through A and E are horizontal. 170. Coefficients for Pier Reactions.—When the spans areequal in length and the load is uniform throughout, the reactionat each support may be found by multiplying the total loadon each span by the coefficients, as given in the followingtable. The Roman numerals represent the number of spansover which the beam is continuous. 268 THEORY OF FRAMED STRUCTURES Chap. VI Reaction Coefficients I i-i 2 2 II 3_iP_38 8 8 III —A. lO lO lO lO IV ii_3^_^_3^_ii28 28 28 28 28 V i5_43_37_37_43_IS38 38 38 38 38 38 S
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